On two statistics of chi-square type based on frequencies of tuples of states of a high-order Markov chain
Маргарита Игоревна Тихомирова, Vladimir Pavlovich Chistyakov · Discrete Mathematics and Applications · 2003
We consider a tuple of states of an ( s – 1)-order Markov chain whose transition probabilities depend on a small part of s – 1 preceding states. We obtain limit distributions of certain χ 2 -statistics X and Y based on frequencies of tuples of states of the Markov chain. For the statistic X , frequencies of tuples of only those states are used on which the transition probabilities depend, and for the statistic Y , frequencies of s -tuples without gaps. The statistical test with statistic X which distinguishes the hypotheses H 1 (a high-order Markov chain) and H 0 (an independent equiprobable sequence) appears to be more powerful than the test with statistic Y . The statistic Z of the Neyman–Pearson test, as well as X , depends only on frequencies of tuples with gaps. The statistics X and Y are calculated without use of distribution parameters under the hypothesis H 1 , and their probabilities of errors of the first and second kinds depend only on the non-centrality parameter, which is a function of transition probabilities. Thus, for these statistics the hypothesis H 1 can be considered as composite.